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More specifically: Let G1 be a subgroup of the group G. Let H1 be a subgroup of the group H. Prove G1 x H1 is a subgroup of G x H.

I've already shown the identity element e is in G1 x H1, and I've already shown closure under the * operator.

However, I don't know how to prove that for every element in G1 x H1, the inverse is an element as well, which I know is the last step to proving what was needed.

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1 Answer

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Let $(g,h) \in G_{1}\times H_{1}$.
Then $g \in G_{1}$ and $h \in H_{1}$.
Then $g^{-1} \in G_{1}$ and $h^{-1} \in H_{1}$ (since they are subgroups, and therefore all elements have inverses).
Therefore $(g^{-1},h^{-1}) \in G_{1}\times H_{1}$.

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